Paid internships are generally above minimum wage. My lowest paid internship was $10/hr, highest was $15/hr. Interns, however, are not legally supposed to be doing production work or run-the-business kind of work. They're there as a learning experience similar to a classroom. They can do project work and attend meetings.
Interns generally aren't classified as workers. They're getting experience in exchange for their time. The employer can't immediately gain from an intern's work. As long as the employer follows the guidelines of the intern regulations, they don't have to follow minimum wage or pay the intern at all (similar to how students don't get paid by their university for attending class).
Your link says: There are some circumstances under which individuals who participate in “for-profit” private sector internships or training programs may do so without compensation.
Which is directly contrary to the point you cite it to support.
Until recently, interns slipped through the cracks somehow. A recent court decision determined that interns actually aren't magically exempt from the law just because they're interns, though, so expect a lot fewer unpaid internships from now on.
Interns generally aren't classified as workers. They're getting experience in exchange for their time. The employer can't immediately gain from an intern's work. As long as the employer follows the guidelines of the intern regulations, they don't have to follow minimum wage or pay the intern at all (similar to how students don't get paid by their university for attending class).